Effectiveness factor at isothermal conditions

For an irreversible first order reaction A → B , at isothermal condition the rate at surface condition can be written as . Then the actual global rate at outer surface condition, .

For a spherical catalyst pellet of radius ‘R_s', mass balance over a spherical shell volume of thickness at steady state will be:

(Rate of diffusion of reactant into element) – (Rate of diffusion of reactant out of element) = Rate of disappearance of reactant within element due to reaction

Rate of reaction per unit mass of catalyst =

Rate of reaction per unit volume of catalyst = .

Where is the density of pellet.

Then the mass balance equation over the element of thickness can be written as

If limit is taken as and assuming that effective diffusivity is independent of concentration of reactant the above equation can be written as

(7)

Boundary conditions:

At the center of the pellet,

(8)

At outer surface

at

(9)

Solving the differential equation (7) with boundary conditions (8) and (9) following relation is obtained

(10)

Where is a dimensionless group defined as